We state and prove a generalization of Kingman’s ergodic theorem on a measure-preserving dynamical system where the μ-almost sure subadditivity condition fn+m ≤ fn + fm◦Tn is relaxed to a μ-almost sure, “gapped,” almost subadditivity condition of the form for some non-negative ρn ∈ L1(dμ) and that are suitably sublinear in n. This generalization has a first application to the existence of specific relative entropies for suitably decoupled measures on one-sided shifts.
We choose the convention that .
See, e.g., Lemma 2 in Ref. 1 for a direct proof. This lemma is to be applied to each of the finitely many integrable functions of the form fq,+ using the fact that σn = o(n).
The criterion on r can be determined as a function of ϵ and the growth of only.
Informally, is “good” because the term appearing in the estimate below can be bounded in terms of f and ϵ directly; is “bad” because the corresponding term fr,+ cannot—but this is rare enough for n ≫ K ≫ 1 in view of step 3.
Since we have assumed in Theorem III.1 that fn,+ is integrable, the integral of fn is well defined in [−∞, ∞), which suffices for Lemma II.1.